Multigraded commutative algebra

多项式方程是数学和科学中最基本的对象之一，它们出现在大量的例子中：费马大定理、运动的物理模型、理想气体定律等等。因此，理解多项式方程的解是一个跨越所有定量领域的基本挑战。PI的工作旨在发展理论和算法两方面的技术，以便更好地理解被赋予额外对称性的多项式方程的解。由于这些方程经常出现在数学或科学应用中，因此这项工作具有广泛影响的潜力。PI的目的是扩大文献的多梯度多项式，建立在他以前的工作，使用虚拟分辨率作为基础的几何应用的多梯度协同。一个项目将提供Hilbert Syzygy定理和Beilinson对角线分解的新多阶类似物，从而建立两个基础结果。第二个项目是关于格林线性协同定理的多重分级版本，它有更直接的几何应用。第三个项目将有计算应用，产生一种新的、可能更快的算法来计算环上同调。该项目还将通过PI对博士生的指导产生更广泛的影响。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Polynomial equations are some of the most fundamental objects in mathematics and science, and they arise in a huge array of examples: Fermat’s Last Theorem, physical models of motion, the ideal gas law, and much more. Understanding the solutions of polynomial equations is thus a fundamental challenge that cuts across all quantitative fields. The PI’s work aims to develop techniques—both theoretical and algorithmic—for better understanding the solutions of polynomial equations that are endowed with extra symmetries. As such equations often arise in mathematical or scientific applications, the work has the potential for wide impact.The PI aims to expand the literature on multigraded polynomials, building on his prior work of using virtual resolutions as a foundation for geometric applications of multigraded syzygies. One project would provide novel multigraded analogues of the Hilbert Syzygy Theorem and of Beilinson's resolution of the diagonal, thus establishing two foundational results. A second project on a multigraded version of Green's Linear Syzygy Theorem would have more direct geometric applications. And a third project would have computational applications, yielding a new and potentially much faster algorithm for computing sheaf cohomology on a toric variety. The project will also have broader impacts through the PI’s mentoring of PhD students.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.